An anecdote won't really prove anything, but I do have experience using multiple FEA tools. I would argue that anyone using that software without a solid understanding of the math behind it is at serious risk of misinterpreting the solutions. There are some easily identifiable failures of these solvers, but there are also quite a few that can give you the wrong answer with no detectable indications unless you comprehend what's going on behind the scenes.
So I think there's definitely a need to teach PDEs and linear algebra. Although, I also agree that the statistics is not emphasized enough. It's a critical element in understanding things like material properties, fatigue, vibration and measurement systems. Those are just as important to ensuring that a design is adequate as getting accurate FE analysis results.